# Coercivity, essential norms, and the Galerkin method for second-kind integral equations on polyhedral and Lipschitz domains

It is well known that, with a particular choice of norm, the classical double-layer potential operator D has essential norm <1/2 as an operator on the natural trace space H^1/2(Γ) whenever Γ is the boundary of a bounded Lipschitz domain. This implies, for the standard second-kind boundary integral equations for the interior and exterior Dirichlet and Neumann problems in potential theory, convergence of the Galerkin method in H^1/2(Γ) for any sequence of finite-dimensional subspaces (ℋ_N)_N=1^∞ that is asymptotically dense in H^1/2(Γ). Long-standing open questions are whether the essential norm is also <1/2 for D as an operator on L^2(Γ) for all Lipschitz Γ in 2-d; or whether, for all Lipschitz Γ in 2-d and 3-d, or at least for the smaller class of Lipschitz polyhedra in 3-d, the weaker condition holds that the operators ±1/2I+D are compact perturbations of coercive operators – this a necessary and sufficient condition for the convergence of the Galerkin method for every sequence of subspaces (ℋ_N)_N=1^∞ that is asymptotically dense in L^2(Γ). We settle these open questions negatively. We give examples of 2-d and 3-d Lipschitz domains with Lipschitz constant equal to one for which the essential norm of D is ≥ 1/2, and examples with Lipschitz constant two for which the operators ±1/2I +D are not coercive plus compact. We also give, for every C>0, examples of Lipschitz polyhedra for which the essential norm is ≥ C and for which λ I+D is not a compact perturbation of a coercive operator for any real or complex λ with |λ|≤ C. Finally, we resolve negatively a related open question in the convergence theory for collocation methods.

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